A NEWTON MULTIGRID METHOD FOR QUASILINEAR PARABOLIC EQUATIONS

A combination of the classical Newton Method and the multigrid method, i.e., a Newton multigrid method is given for solving quasilinear parabolic equations discretized by finite elements. The convergence of the algorithm is obtained for only one step Newton iteration per level. The asymptotically computational cost for quasilinear parabolic problems is O（NNk） similar to multigrid method for linear parabolic problems.     

BLOW-UP SOLUTIONS FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS WITH MIXED BOUNDARY CONDITIONS

The type of problem under consideration is where D is a smooth bounded domain of RN, q = |(?)u|2. By constructing an auxiliary function and using Hopf's maximum principles on it, existence theorems of blow-up solutions, upper bound of "blow-up time" and upper estimates of "blow-up rate" are given under suitable assumptions on a, b, f, g, σand initial date U0(X). The obtained results are applied to some examples in which a, b, f, g and σare power functions or exponential functions.     

QUASILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS WITH DISCONTINUOUS NONLINEARITIES

1.IntroductionInthispaperweshallconsideradiscontinuousnonlinearnonmonotoneellipticboundaryvalueproblem,i.e.aquasilinearelliptichemivariationalinequalitywhereAn=--Z:=,hat(x,u,7u) ac(x,usac)an4thesymbolaldesignatesclarke'sgeneralizedgradientofalocallyL...     

THE DIRICHLET PROBLEM AND NEUMANN PROBLEM FOR NONLINEAR PARABOLIC SYSTEMS OF SECOND ORDER EQUATIONS WITH MEASURABLE COEFFICIENTS IN A HIGHER DIMENSIONAL DOMAIN

1 Formulation of Dirichlet Problem and Neumann Problem fOrParabolic SystemsLet O be a bounded domain in Rn and the boundary On E C2. Denote Q = g x I,I =0 < t 5 T, 0 < T < ool 0Q = Sl U S2 is the parabolic boundary where S1 = a x {t = 0} is thebottom and S2 = an x I is the lateral boundary. We consider the nonlinear parabolic systemof second order equationsFh(t, xl ut D.u, D:u) -- Hukt = 0 in Q, k = 1, 2,'',m. (l.l)Under certain conditions, system (1.1) can be reduced to the fOrmw…     

SOME STABILITY THEOREMS FOR DIFFERENCE EQUATIONS WITH TIME DELAY

1.IntroductionDifferellceequationswithtimedelayhavebeenanimportantclassofmathematicallllodels.Fol'instance,inbiology,systemsofdifferenceequationshavebeedmuchusedtomodeltheinteractiollsofspecieswithlion-overlappinggeneration,butthemodelwillberelatedtodifferenceequationswithtimedelaywhenollemustconsidertheinfluenceoftimedelayeffectforspecies11limber.Thereisextensiveresearchondifferentialequationswithtimedelaybutthereisoillylittleworkdoneondifferenceequationswithtimedelay.Ouraimistoestablishstabi…     

A CLASS OF SINGULARLY PERTURBEDBOUNDARY VALUE PROBLEMS FOR NONLINEARDIFFERENTIAL SYSTEMS

Considerthesingularlyperturbednonlinearproblemfory=(y1,…,yn)oftheformwhereεisasmallpositiveparameter,f=(f1,…,fn),g=(g1,…gn),fi(x,y')≡fi(x,y'1,…,y'n),gi(x,y')≡gi(x,y'1,…,y'n),A=(A1,…,An),B=(B1,…,Bn).ManyauthorssuchasHowes[1],O'DonnelllZ],Lin[3],Mo[4--7]andothershavediscussedthisproblemundervarioussituationsusingthemethodofdifferentialinequalities.Thispaperconcernsaclassofboundaryvalueproblemsfornonlineardifferentialsystemsusingtheiterationmethodandthetheoryofdifferentialinequaliti…     

THE SINGULAR PERTURBATION OF NONLOCAL BOUNDARY VALUE PROBLEMS FOR NONLINEAR ELLIPTIC EQUATION OF FOURTH ORDER

Chenandothersstudiedaclassofsingularlyperturbedboundaryproblems[1--3].NowweconsiderthefollowingsingUlarlyperturbednonlocalboundaryvalueproblemforthenonlinearellipticequationoffourthorder.whereEisapositiveparameter,bistheLaplacian,ndenotesaboundedregi...     

BOUNDARY SOLUTIONS OF OPERATOR EQUATIONS IN PARIAL ORDERING SETS WITH APPLICATIONS TO SYSTEMS OF FUNCTIONAL

1.IntroductionThepaPers[l-5]studythecoupledfixedp0intsofmixedmonotoneoperatorsinBanachspaces;theyshowusgeneralizationsofthefixedpoillttheoremsonincreasingoperatorsanddecreasingoperators.Applicationsoftheseresultsmaybefoundinmanyfieldsofmathematicssuchasnonlineaxdifferentialequations,illtegralequationsanddynamicprogrammingtheories.Inthispaper,westudytheekistenceofsolutionsandboundarysolutionstoasystem0foperatorequationsinpartialorderingsets.Wegeneralizeandimproveonsomeresultsaboutthecoupledfix…     

A PRIOR ESTIMATE AND EXISTENCE OF POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATION WITH THE THIRD BOUNDARY VALUE PROBLEM

1 IntroductionLet n be a bounded open domain with boundary aa, n c R", N 3 2. We consider a priorestimate and ekistence of positive solution of the fOllowing equationsWhere operatorL conforms to the Sum Ru1es, that is: for a repeated i or j, it mean to sum them up from 1 toN, ai;(x), ai(x), a(x) are bounded measurable in a. FOr operator L, we alWays assume: thereealst positive constants po and U1) such thatf(u) is a nonlinear function which grows in the form of Up with p 2 1, n is the out…     

ASYMPTOTIC STABILITY OF SOLUTION TO THE INITIAL-BOUNDARY VALUE PROBLEM FOR SCALAR VISCOUS CONSERVATION LAWS CORRESPONDING TO RAREFACTION WAVES

1.IntroductionWeconsidertheinitial-boundaryvalueproblem(IBVP)forscalarviscousconservationlaw(ut f(u).=uxx,xE(0, co)=R ,t>0,o,(n \--9,ifI~9,f~M(IBVP){u(0't)=u~(t)-1~i---t:7),[u(x,0)=u000={=:0)f:LO .,wheref"(u)>0foralluunderconsideration,andu--u (1.1),andu~(t)~u~EL'(R ),U'(t)EL'(R ).(1.2)Undercondition(1.l)or(l.l)',thecorrespondingRiemannproblem{;;ot:<:>;i,>=x{,:3'f:,:(13)yieldstherarefaCtionwavesolution(u~)x5f'(u~)t,r"(X,t)=17j,)~1(…     

ASYMPTOTIC BEHAVIOR OF THE SOLUTION FOR THE NEUTRON TRANSPORT EQUATION IN A SLAB WITH GENERALIZED REFLECTING BOUNDARY CONDITIONS

This paper discusses the neutron transport equation in a slab with generalized reflectingboundary conditions.By means of the positive C_0-semigroup theory,we have proved that this problemhas a unique nonnegative solution and found the spectral property of the corresponding transportoperator.Finally we give the asymptotic behavior of the solution for this equation.     

ASYMPTOTIC STABILITY OF VISCOUS SHOCK PROFILE FOR NON-CONVEX SYSTEM OF ONE-DIMENSIONAL VISCOELASTIC MATERIALS WITH BOUNDARY EFFECT

1 IntroductionWe consider the system of one-dimensiona1 viscoelastic materials with non--convekity of thefOrm.with the illitial-boundary conditionshere v is the strain, u is the ve1ocity p > 0 is the viscous constallt, v and u are givenconstants, a is the pressure to be a known smooth fUnction of v satisfying the fOllOwing signconditions:a'(v) > 0 fOr all v u-nder consideration, (1.4)a"(v) 2 0 fOr v 2 0 under consideration, (l.5)so that rr(v) has a point of inflection at v = 0. We see that…     

ON ASYMPTOTIC FORMULAE OF EIGENVALUE COUNTING FUNCTIONS OF LAPLACIAN ON AN UNBOUNDED DOMAIN WITH A FRACAL BOUNDARY

1. IntroductionLet D be an unbounded open set in R"(n 2 2) with finite Lebesgue measure. We considerthe following eigenvalue problem on a.where A = Z:=, 6 denotes the Laplacian on a and r ~ oa, the boundary of g. In thispaper we have a special interest in r which is considerably irregular like a fractal.Let N(A) denote the eigenvalue counting function of (P): the number of eigenvalues notexceeding A. When n is a bounded open set, it is well known that Weyl's classical asymptoticformulae f…     

ASYMPTOTIC EXPANSION AND ASYMPTOTIC BEHAVIOR OF THE SOLUTION FOR THE TIME-DEPENDENT NEUTRON TRANSPORT PROBLEM IN A SLAB WITH GENERALIZED BOUNDARY CONDITIONS

In this paper,the time-dependent neutron transport integro-differential equationin a nonuniform slab with generalized boundary conditions and initial value is considered forgeneral cases concerned with an arbitrary nonhomogeneous medium possibly with cavity,withthe anisotropic scattering and fission,and with continuous energy varying from null to any finiteconstant or from one positive constant to another positive constant.We prove that the correspon-ding neutron transport operator A has finite Spectrum points in any strip {λ|β_1≤R(?)λ≤β_2}whereβ_2>β1>-λ~*(λ~* is the essential infimum of v∑(x,v)),and obtain the asymptotic expansion ofthe time-dependent solution which exists and is unique.Furthermore,we give the existence ofthe dominant eigenvalue and indicate the asymptotic behavior of the neutron density as t→+∞.     

A PRECONDITIONER FOR THREE－DIMENSIONAL DOMAIN DECOMPOSITION METHODS WITH LAGRANGE MULTIPLIERS

In this paper we consider domain decomposition methods for three-dimensional elliptic problems with Lagrange multipliers, and construct a kind of simple preconditioner for the corresponding interface equation. It will be shown that condition number of the resulting preconditioned interface matrix is almost optimal.